Were there any similarities between the Puritans and Quakers?
There was very little in the way of similarities between the Puritans and the Quakers in Colonial America. In fact, the Puritans were sufficiently opposed to the Quakers that there were instances of Quaker missionaries being executed by Puritans.
The major reason for this was that the Quakers were much less dogmatic than the Puritans. They believed that all people had the "Inner Light" of God within them and that listening to this "Inner Light" was much important than reading scripture or listening to what preachers had to say. The Puritans wanted a rigidly controlled society in which all people worshipped God in the same way. They felt that this was necessary to prevent God from punishing their community.
The Puritans were much more controlling and dogmatic than the Quakers. Therefore, they were very different in both religious doctrine and social habits. The only real similarity is that both were Christian sects who went to the American colonies to try to have more of an ability to worship God in their own way.